interval notation and sin

Question

Solve with the methods shown in this section exactly on the interval [0, 2𝜋).

 (Enter your answers as a comma-separated list.)

15 sin(2t) + 25 sin(t) = 0

Answer 1

The double angle identity for sine shows that sin(2t) = 2sin(t)cos(t), so the original equation can be written as:

15(2)sin(t)cos(t) + 25sin(t) = 0

The GCF if the terms on the left is 5sin(t) so factor that out:

5sin(t)[6cos(t) + 5] = 0

Now using the zero product property one of those factors must equal 0, so:

sin(t) = 0 OR 6cos(t) + 5 = 0.

The first equation results in:

t = arcsin(0) = 0, π

The 2nd equation results in:

t = arccos(-5/6). A calculator returns a value (in radians) equal to about 2.556. There is another angle in the 3rd quadrant that also has a cosine equal to -5/6 (2π-2.556 ≈ 3.727)

You can also graph the function f(x) = 15sin(2x) + 25sin(x) and determine its x -intercepts:

Check it out:

desmos.com/calculator/figfwonjlb